On commuting pairs in arbitrary sets of 2x2 matrices

Let ${\text{Mat}}_2(\mathbb{R})$ be the set of $2\times 2$ matrices with real entries. For any $\epsilon >0$ and any finitely–supported probability measure $\mu$ on ${\text{Mat}}_2(\mathbb{R})$, we prove that either $$T(\mu )=\sum _{X,Y\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu ),XY=YX}\mu (X)\mu (Y)<\epsilon$$ or there exists some finite set $S$ contained in a $2$-dimensional subspace of ${\text{Mat}}_2(\mathbb{R})$ such that $\mu (S)\ge \epsilon \mathrm{/}8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when $$\mu ((a_{i,j}{)}_{1\le i,j\le 2})=\nu (a_{1,1})\dots \nu (a_{2,2})\text{ for every }{a}_{1,1},\dots ,a_{2,2}\in \mathbb{R},$$ with $\nu$ being some finitely–supported probability measure on $\mathbb{R}$. For instance, when $A\subset \mathbb{R}$ is a generalised arithmetic progression or multiplicative progression of dimension $d$ and $\nu =1_A\mathrm{/}\mathrm{\mid }A\mathrm{\mid }$, our techniques imply that $\mathrm{\mid }A{\mathrm{\mid }}^{-3}{\ll }_{d}T(\mu ){\ll }_d\mathrm{\mid }A{\mathrm{\mid }}^{-3}$. Our methods highlight the connections of this problem to results in incidence geometry, growth in groups phenomenon as well as Bourgain–Chang type sum-product estimates over $\mathbb{R}$. The latter includes applications of Schmidt’s subspace theorem and the resolution of the weak polynomial Freiman–Ruzsa conjecture over integers.